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Experimental evidence of self-localized and propagating spin wave modes in obliquely
magnetized current-driven nanocontacts
Stefano Bonetti1,∗ Vasil Tiberkevich2, Giancarlo Consolo3,4, Giovanni Finocchio4,
Pranaba Muduli5, Fred Mancoff6, Andrei Slavin2, and Johan A˚kerman1,5
1 Materials Physics, Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden
2 Department of Physics, Oakland University, Rochester, MI 48309 USA
3 Department of Physics, University of Ferrara, Ferrara, Italy
4 Department of Matter Physics and Electronic Engineering, University of Messina, Messina, Italy
5 Department of Physics, University of Gothenburg, Gothenburg, Sweden
6 Everspin Technologies, Inc., 1300 N. Alma School Road, Chandler AZ, USA
Through detailed experimental studies of the angular dependence of spin wave excitations in
nanocontact-based spin-torque oscillators, we demonstrate that two distinct spin wave modes can
be excited, with different frequency, threshold currents and frequency tuneability. Using analytical
theory and micromagnetic simulations we identify one mode as an exchange-dominated propagating
spin wave, and the other as a self-localized nonlinear spin wave bullet. Wavelet-based analysis of the
simulations indicates that the apparent simultaneous excitation of both modes results from rapid
mode hopping induced by the Oersted field.
Spin-polarized currents passing through a thin mag-
netic film can excite spin waves via the spin-transfer-
torque effect [1, 2]. In his pioneering paper [3],
Slonczewski predicted that such spin waves, excited
in perpendicularly magnetized free layer underneath a
nanocontact [4], would be exchange-dominated, propa-
gating radially from the nanocontact region, with a wave
number k inversely proportional to the nanocontact ra-
dius Rc (k ' 1.2/Rc). Rippard et al. [5] subsequently
demonstrated that, while Slonczewski’s theory correctly
describes the frequency and threshold current of spin
waves excited in perpendicularly magnetized films, it fails
to describe spin waves excited when the same film is
magnetized in the plane. It was later shown theoreti-
cally that in the case of in-plane magnetization, apart
from the Slonczewski-like propagating spin wave mode,
it is possible to excite a self-localized nonlinear spin wave
mode of solitonic character – a so-called standing spin
wave bullet [6]. While the current-induced excitation of
the spin wave bullet was subsequently confirmed in sev-
eral numerical simulations [7–10], and angular dependent
measurements have been presented in the literature [11],
no clear experimental evidence of a spin wave bullet nor a
characterization of its properties, has yet been presented.
In this Letter, we study the angular dependence of spin
wave excitations in nanocontact based spin-torque oscil-
lators (STO), and demonstrate that when the free layer is
magnetized at sufficiently small angles θe . 55◦ w.r.t the
plane, two distinct and qualitatively different spin wave
modes can be excited by the current passing through the
nanocontact. The two modes have different frequencies,
f , different threshold currents, Ith, and opposite sign
of the frequency tuneability, df/dI. Through compari-
son with theory [6, 12] and micromagnetic simulations,
we show that the higher-frequency, blue-shifting mode
is well described by the Slonczewski propagating mode,
and that the lower-frequency, red-shifting mode exhibits
all the predicted properties of a localized spin wave bul-
let. Using time-frequency wavelet-based analysis of our
micromagnetic simulations we furthermore demonstrate
that the apparent simultaneous excitation of both modes,
as observed in our frequency domain experiments, results
from non-stationary switching between the two modes on
the sub-ns time scale.
The magnetically active part of the sample is a
Co81Fe19(20 nm)/Cu(6 nm)/Ni80Fe20(4.5 nm) thin film
tri-layer, patterned into a 8 µm × 26 µm mesa. On top
of this mesa, a circular Al nanocontact having nominal
diameter 2Rc = 40 nm was defined through SiO2 using
e-beam lithography (see Ref. [13] for details). An exter-
nal magnetic field of constant magnitude (µ0He = 1.1 T)
was applied to the sample at an angle θe w.r.t the film
plane. Details of the measurements setup are given in
Ref. [14]. The excited spin waves modulate the magne-
toresistance of the device and are detected as a microwave
voltage signal. Microwave excitations were only observed
for a single current polarity, corresponding to electrons
flowing from the “free” (thin NiFe) to the “fixed” (thick
CoFe) magnetic layer. All measurements were performed
at room temperature. While the results presented here
all come from a single sample, we have confirmed that the
results obtained on several other devices are qualitatively
similar.
Figure 1 shows the angular dependence of the mi-
crowave frequencies generated at a constant current of
I = 14 mA. The most striking feature is the existence
of two distinct signals for sufficiently small values of θe.
The frequencies of these two signals differ by about 2.5
GHz at angles up to θe = 40◦, and then start to ap-
proach each other up to a critical angle θe = θc ≈ 55◦
where the lower frequency signal disappears. The general
behavior of the two signals remains the same at higher
currents (I = 18 mA), with a slight increase of both θc
and the frequency separation to about 58◦ and 3 GHz

20 15 30 45 60 75 90
0
10
20
30
40
Applied field angle θ
e
[deg]

f
[G
Hz
]


Propagating mode
Spin wave bullet
0 15 30 45 60 75 90
0
10
20
30
40
Applied field angle θ
e
[deg]

f
[G
Hz
]
Figure 1. (Color online) Measured frequencies of the observed
spin wave modes as a function of the applied field angle θe at
I = 14 mA and µ0He = 1.1 T. Inset: theoretically calculated
frequencies of the propagating (upper curve) and the bullet
(lower curve) modes at the current threshold, for nominal
parameters of the nanocontact STO.
respectively.
Fig. 2 shows the angular dependence of the threshold
current Ith for both signals. Ith is found from peak power
(p) measurements in the sub-critical regime (I < Ith) an-
alyzed using the method proposed in Refs. [15, 16], and
employed in Ref. [17]. Since 1/p(I) ∼ (Ith − I) in this
regime, Ith for each signal can be directly determined
from the intercept of a straight line through 1/p(I) vs. I
with the current axis. We extracted Ith only for the mag-
netization angles 20◦ < θe < 80◦, since outside this range
the signal was too weak to allow for a reliable analysis.
We note that the low-frequency signal always has the
lower threshold current (within the noise of the analysis),
in particular at low field angles. As the angle increases,
the Ith values for the two signals gradually approach each
other and become essentially equal close to θc. For the
lower-frequency signal, the data is plotted up to θe = 47◦,
since above this angle the signal is too low to allow for a
reliable determination of Ith.
The upper inset in Fig. 2 shows the experimental tune-
abilities df/dI of the two signals. The lower-frequency
signal always red shifts with current (df/dI < 0) with val-
ues ranging from -40 to -110 MHz/mA. In contrast, the
higher-frequency signal blue shifts with current (df/dI >
0) with values ranging from +50 to +150 MHz/mA
for θe < 45◦ and from +300 to +400 MHz/mA when
θe > 55◦. The opposite tuneability sign can be clearly
seen in Fig. 3a-b where the microwave power of both sig-
nals is color mapped onto the frequency-current plane.
At θe = 30◦ (Fig. 3a)) both signals are visible with a
lower Ith and a clear red shift for the lower-frequency
signal, and a higher Ith and a clear blue shift for the
0 15 30 45 60 75 90
0
5
10
15
20
Applied field angle θ
e
[deg]
Th
re
sh
ol
d
cu
rre
nt
I
th


[m
A]
0 15 30 45 60 75
−200
0
200
400
θ
e
[deg]

df
/d
I
[M
Hz
/m
A]
15 30 45 60 75
5
10
15
20
θ
e
[deg]

I th


[m
A]
Figure 2. (Color online) Measured threshold current for the
propagating (empty triangles) and the bullet (filled circles)
modes as a function of applied field angle θe. Lower inset:
theoretical threshold current vs applied field angle. Upper
inset: df/dI vs. θe for the propagating (empty triangles) and
the bullet (filled circles) modes as a function of θe. Filled
squares are the results of micromagnetic simulations.
higher-frequency signal. At a larger magnetization an-
gle, θe = 65◦ (Fig. 3b)), only the higher-frequency, blue-
shifting signal is visible.
In order to gain a physical understanding of the ex-
perimental observations, we use the theoretical model
of spin wave excitations developed in Refs. [6, 12]. In
this model, one of the excited modes is directly related
to Slonczewski’s propagating spin wave mode for a per-
pendicularly magnetized nanocontact STO [3], but now
generalized for the case of an oblique orientation of the
free layer magnetization. The mode retains its propa-
gating character, i.e. it carries energy away from the
nanocontact area, and its threshold current is approxi-
mately given by:
Ipropth ≈
[
Γ(θe) + 1.86D(θe)/R2c
]
/σ(θe) , (1)
where Γ(θe) is the Gilbert damping rate in the free layer,
D(θe) is the spin wave dispersion coefficient, Rc is the
nano-contact radius, and σ(θe) is the spin-polarization
pre-factor (see Eq. (2) in Ref. [6]). The second term in
Eq. (1) is independent of the spin wave damping Γ and
describes radiative energy losses due to the propagating
character of the mode. It is usually larger than the first
term, which describes losses due to direct energy dissi-
pation in the nanocontact area. The frequency of the
propagating mode
ωprop ≈ ω0(θe) + 1.44D(θe)/R2c +N(θe)|a|2 (2)
is hence typically higher than the FMR frequency ω0 of
the free layer, and depending on the sign of the nonlinear

3Figure 3. (Color online) Comparison of experiment with mi-
cromagnetic simulations: (a, b) Measured microwave power,
presented as color maps onto the frequency-current plane, for
two applied field angles (a) θe = 30◦ and (b) θe = 65◦, where
filled symbols show the results of the micromagnetic simula-
tions. (c) Wavelet analysis, and (d) Fast Fourier transform,
of the micromagnetic simulation at θe = 30◦ and I = 16 mA.
frequency shift, N = ∂ω/∂|a0|2, either blue shifts or red
shifts with increasing current.
The model also predicts the existence of an additional
mode whose amplitude and spatial extension are stabi-
lized by two concurring processes: i) energy dissipation
due to Gilbert damping and ii) energy gain from the
spin-polarized current. This mode is self-localized, has a
two-dimensional solitonic character with a bullet shaped
amplitude profile (hence also called a spin wave bullet),
and does not carry any energy away from the nanocon-
tact area. The expression for its threshold current con-
sequently lacks the radiative term, and, therefore, Ibulth is
directly proportional to the spin wave damping:
Ibulth ≈ βΓ(θe)/σ(θe) , (3)
where the dimensionless coefficient β ∼ 1 depends on the
parameters of the system (for further details see [6]). As
a consequence, the spin wave bullet mode always has a
lower threshold current than the Slonczewski-like prop-
agating mode, except very close to θcr where β diverges.
The spin wave bullet mode does not exist at all mag-
netization angles. Only for N < 0 can the nonlinearity
counteract the dispersion-related spreading of the spin
wave profile. Since N is a function of the magnetiza-
tion angle, and changes sign from always negative for an
in-plane magnetization to always positive for a perpen-
dicular magnetization [16], the spin wave bullet mode
only exists for applied field angles smaller than a certain
critical value θcr, and does not exist for θe > θcr [12]. As
a direct consequence of the negative N, the frequency of
the spin wave bullet mode
ωbul ≈ ω0(θe) +N(θe)|a0|2 (4)
always lies below the FMR frequency ω0, and continues
to decrease with increasing current (it always red shifts).
The inset of Fig. 1 shows the calculated f vs. θe for
the two modes, using the nominal parameters of the mea-
sured STO. The experimental data confirms the theoret-
ical predictions, such as the existence of a critical an-
gle below which a lower-frequency mode is excited, and
a qualitatively similar angular dependence of the fre-
quency. The inset of Fig. 2 shows Ith vs. θe for both
modes. Again, the experimental data exhibits the same
qualitative behavior as predicted by theory. In particu-
lar, Ith of the lower-frequency mode is always lower than
that of the higher-frequency mode. The combined quali-
tative agreement provides a strong argument for identify-
ing the observed higher-frequency mode as a propagating
mode, and the lower-frequency mode as a spin wave bul-
let.
There are, however, some notable differences between
theory and our experiments, the most striking being the
apparent simultaneous excitation of both modes. Such a
co-excitation is neither supported by theory [6, 12], nor
by micromagnetic simulations [7, 9], where, on the con-
trary, a hysteresis between the two modes was observed.
Another significant difference is the lack of any observed
red shift of the propagating mode. While our experi-
ments clearly demonstrate that the propagating mode is
blue shifted at all field angles, Eq. (2) predicts a red
shift also for this mode when N < 0, i.e. for θe < θcr, in
clear contradiction with our experimental data. As we
show below, both effects can be explained by taking into
account the large Oersted field generated in the nanocon-
tact, which was previously ignored in [6, 7, 9, 12], but can
be accounted for in micromagnetic simulations.
Micromagnetic simulations were carried out in a box
shaped free layer volume 800× 800× 4.5 nm3 with con-
stant cell size 4× 4× 4.5 nm3. A uniform spin polarized
current acted on a quasi-cylindrical sub-volume of the
free layer with an adjustable radius Rc. The material
properties of the NiFe free layer were: saturation mag-
netization µ0MS,free = 0.7 T, Gilbert damping constant
αG = 0.01, exchange constant A = 1.1× 10−11 J/m, and
spin-torque efficiency  = 0.3. The thickness and satu-
ration magnetization of the CoFe fixed layer were 20 nm
and µ0MS,fixed = 1.8 T, respectively, and a minimization
of the fixed layer magnetostatic energy in the applied
field determined the fixed layer magnetization angle and
consequently the polarization angle of the spin polarized
current. No magneto-crystalline anisotropy, RKKY in-
teraction, or dipolar coupling between the two magnetic
layers were taken into account. The external field mag-
nitude was fixed at µ0Hext = 1.15 T, and its direction
was varied to fit the experimental data. All simulations
were done at T = 0 K.

4The results of our micromagnetic simulations are
shown as symbols in Fig. 3a-b after optimization of
Rc = 32 nm and magnetic field angles of 35◦ and 70◦
respectively. The quantitative agreement with the exper-
imental data is remarkable: excitation frequency, range
of the mode existence, and frequency tuneability of the
two modes are well reproduced by the simulations. The
reason for the larger simulated contact radius (32 nm
vs. nominally 20 nm in the experiment) is likely due to
two neglected effects: i) current crowding at the contact
perimeter, essentially increasing the effective contact ra-
dius compared to a uniform current, and ii) lateral cur-
rent spread in the free and fixed layers. The simulated
field angles agrees with those used in the experiments to
within 10%.
It is clear from our simulations that the inclusion of the
Oersted field affects the magnetization dynamics in sev-
eral significant ways. First, the Oersted field has a strong
qualitative impact on df/dI of the propagating mode,
making it positive at all investigated field angles, effec-
tively resolving the apparent discrepancy between our
experimental observations and theory. df/dI of the bul-
let mode, on the other hand, remains largely unchanged
with the inclusion of the Oersted field. If we now add
the simulated df/dI at 35◦ and 70◦ to the upper inset of
Fig. 2, we also find a remarkable quantitative agreement
with the experimental values for both modes.
Second, the inclusion of the Oersted field also makes
the simulations reproduce the apparent simultaneous ex-
citation of both modes. While our experimental setup
is limited to measurements in the frequency domain,
our micromagnetic simulations allow us to also investi-
gate the detailed temporal evolution of the instantaneous
power in each mode. Fig. 3c shows the result of a wavelet
analysis [18] of the micromagnetic data, where the dashed
line indicates the instantaneous maximum power as a
function of time. The analysis makes it clear that the two
modes are in fact never excited at the same time. The
instantaneous microwave power instead exhibits a persis-
tent (at T = 0 K also periodic) hopping between the two
modes with a very high hopping frequency exceeding 1.5
GHz. It is noteworthy that we only observe such hop-
ping when the Oersted field is properly included, which
is likely related to the strong spatial inhomogeneities it
induces in the vicinity of the nanocontact [19, 20].
Finally, the influence of the Oersted field also explains
the large quantitative discrepancy between the analyti-
cally calculated Ith and the experimental value (Fig. 2).
Previous micromagnetic simulations have demonstrated
that the Oersted field can cause a substantial (up to four-
fold) increase of Ith (see Fig. 2 in Ref. [20]), which hence
agrees much better with our experimental data.
In conclusion, we have presented a detailed experi-
mental study of the field angle dependence of spin wave
excitations in nanocontact based spin-torque oscillators.
We find that two distinct and qualitatively very different
spin wave modes can be excited for applied field angles
θe . 55◦. Through a comparison of our experimental
measurements of three different fundamental properties
(f , df/dI, and Ith) with both previously developed ana-
lytic theories [3, 6, 12] and our own micromagnetic simu-
lations, we unambiguously identify the higher-frequency
mode as an exchange-dominated propagating spin wave,
and the lower-frequency mode as a self-localized non-
propagating solitonic mode, i.e. a spin wave bullet. Our
micromagnetic simulations show that not only is the Oer-
sted field required to explain the sign of df/dI, and the
magnitude of Ith, it is also solely responsible for the rapid
(sub-ns) hopping between the two modes, which in fre-
quency domain experiments make them appear as simul-
taneously excited.
We gratefully acknowledge financial support from The
Swedish Foundation for strategic Research (SSF), the
Swedish Research Council (VR), the Go¨ran Gustafsson
Foundation, the Knut and Alice Wallenberg Founda-
tion, by the Contract no. W56HZV-09-P-L564 from the
U.S. Army TARDEC and RDECOM, by the Grant no.
ECCS-0653901 from the National Science Foundation of
the USA, and by the Oakland University Foundation.
Johan A˚kerman is a Royal Swedish Academy of Sciences
Research Fellow supported by a grant from the Knut and
Alice Wallenberg Foundation.
∗ bonetti@kth.se
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